Let $f(x)$ be a third-degree polynomial with real coefficients satisfying \[|f(1)|=|f(2)|=|f(3)|=|f(5)|=|f(6)|=|f(7)|=12.\]Find $|f(0)|$.
Each of the six values $f(1),$ $f(2),$ $f(3),$ $f(5),$ $f(6),$ $f(7)$ is equal to 12 or $-12.$  The equation $f(x) = 12$ has at most three roots, and the equation $f(x) = -12$ has at most three roots, so exactly three of the values are equal to 12, and the other three are equal to $-12.$

Furthermore, let $s$ be the sum of the $x$ that such that $f(x) = 12.$  Then by Vieta's formulas, the sum of the $x$ such that $f(x) = -12$ is also equal to $s.$  (The polynomials $f(x) - 12$ and $f(x) + 12$ only differ in the constant term.)  Hence,
\[2s = 1 + 2 + 3 + 5 + 6 + 7 = 24,\]so $s = 12.$

The only ways to get three numbers from $\{1, 2, 3, 5, 6, 7\}$ to add up to 12 are $1 + 5 + 6$ and $2 + 3 + 7.$  Without loss of generality, assume that $f(1) = f(5) = f(6) = -12$ and $f(2) = f(3) = f(7) = 12.$

Let $g(x) = f(x) + 12.$  Then $g(x)$ is a cubic polynomial, and $g(1) = g(5) = g(6) = 0,$ so
\[g(x) = c(x - 1)(x - 5)(x - 6)\]for some constant $c.$  Also, $g(2) = 24,$ so
\[24 = c(2 - 1)(2 - 5)(2 - 6).\]This leads to $c = 2.$  Then $g(x) = 2(x - 1)(x - 5)(x - 6),$ so
\[f(x) = 2(x - 1)(x - 5)(x - 6) - 12.\]In particular, $|f(0)| = \boxed{72}.$